We first stated the quadrupole correction to the energy to be the dot product of two rank 5 tensors, with 5 elements. Later however it is defined as the dot product of two matrices with 9 elements. How does this work? Were the matrix elements used adapted to this?
Hey Jannes, here’s my take on this. What I understood was that the dot product is some sort of generalisation of a vector dot product in say R^n: pointwise multiplication and add all the results. So if the tensors of rank 2 (I think you meant rank 2 when you wrote rank 5?) can be represented by 3 x 3 matrices, hence 9 elements for which you do this dot product procedure. The conditions symmetric and traceless reduce the number of independent components to 5. You can build the entire matrix from these 5, and hence there is a way to compute the dot product when only given these 5 numbers. Hope this is all correct and clear.